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Nonstandard analysis (Spring 2018)

MATH 649B Nonstandard Analysis 

Spring 2018

Day and Time: MWF 1:30-2:20, Keller 314
Professor: David Ross, PSB 319, ross@math.hawaii.edu

Nonstandard analysis is the art of making infinite sets finite by ex-
tending them.

– Michael Richter

Description:
Nonstandard analysis is an area of mathematics which lives at the interface of mathematical logic (especially model theory) and classical mathematics. Originally introduced as a way to make it possible to work with infinitesimals in a rigorous fashion, it has developed into a powerful methodology with applications in many areas of mathematics. It is especially useful when the concept of limit is central, or when an infinitary or continuous situation has a natural discrete or combinatorial intuition.

This course will be an introduction to the subject, with an eye to getting into ‘real’ applications as quickly as possible. The choice of applications will depend a bit on the background and interests of the students in the class, and might include a bit of topology, probability, measure theory, geometric groups, and additive number theory.

Prerequisites:
The course will be entirely self-contained, though mathematical training and experience at the early graduate level is assumed. In particular, some background in the applications areas won’t hurt.

Text:
The course will mainly be out of lecture notes.

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Real geometry (Spring 2018)

Real geometry, from algebraic to subanalytic and beyond.
MATH 649K, Spring 2018

We will study real algebraic sets (with emphasis on geometry over algebra), semialgebraic sets (sets
defined by polynomial equalities and inequalities) , and subanalytic sets. We will consider the basic
properties of these sets, study of appropriate classes of functions on them, topology, applications.
We may get into further classes of “tame geometry” such as o-minimal geometries.
There are no particular prerequisites, just a certain level of mathematical maturity; undergraduate
mathematical analysis and topology should be enough.

Instructor: Les Wilson

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Representation theory (Spring 2018)

Math 649: Representation theory.

This course is intended to become a regular offering in the ‘algebra and number theory’ collection of courses and so is not really a 649; rather it is the official ‘algebra and number theory’ offering this Spring. Here is a list of topics that are planned for this course.

Linear representations of finite groups: basic theory, group algebras, Schur’s lemma, Maschke’s theorem, the regular representation, character theory, induced representations, Frobenius reciprocity, examples, applications.

Representation theory of semisimple Lie algebras: the basics of Lie algebras, representations of $\mathrm{sl}_2(\mathbb C)$, representations of higher rank semisimple Lie algebras, highest weight theory, roots, relation to the representation theory of semisimple Lie groups.

Further topics may include (some but not all of): in-depth examples such as the symmetric group or finite linear groups, Burnside’s theorem, applications to combinatorics, harmonic analysis on finite groups, modular representation theory, representation theory of compact groups, the classification of complex semisimple Lie algebras and Dynkin diagrams.

Textbook: Fulton and Harris’ “Representation theory, a first course”, supplemented with other texts as necessary.
Instructor: Robert Harron

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Student learning outcomes for graduate programs

Student Learning Outcomes in the PhD Program:
1. Mastery of graduate level mathematics in core areas including at least 2 of analysis, algebra, topology, applied mathematics.
2. Familiarity with the breadth of modern mathematics, by successful completion of a range of advanced courses.
3. Deep knowledge of a specific area of specialization.
4. Ability to accomplish significant mathematical research.
5. Ability to write professional quality mathematics.
6. Ability to present advanced research mathematics to a mathematics audience.
7. Ability to learn advanced mathematics independently.

Student Learning Outcomes in the MA Program:
1. Mastery of graduate level mathematics in core areas including at least 3 of analysis, algebra, topology, applied mathematics.
2. Familiarity with the breadth of modern mathematics, by successful completion of a range of advanced courses.
3. Ability to engage in an independent mathematical project.
4. Ability to communicate mathematics effectively in writing.
5. Ability to present advanced mathematical ideas to a mathematics audience.